Weak sufficiency of quantum statistics

Vol. 60 (2007) REPORTS ON MATHEMATICAL PHYSICS No. 3

WEAK SUFFICIENCY OF QUANTUM STATISTICS

KATARZYNA L U B N A U E R , A N D R Z E J L U C Z A K , H A N N A PODSI~DKOWSKA

Faculty of Mathematics and Computer Science, L6d~ University, ul. S. Banacha 22, 90-238 L6di, Poland (e-mails: [email protected], [email protected], [email protected])

(Received October 23, 2006 - Revised October 4, 2007)

For a quantum observable T and a family of vector states in a separable Hilbert space we give necessary and sufficient conditions in order that T be a weakly sufficient statistic for this family. The definition of weak sufficiency we employ is a natural generalisation of the classical one based on the factorization criterion for a family of distributions dominated by a a-finite measure.

1991 Mathematics Subject Classification: Primary: 46L53; Secondary: 81S05.

Keywords: quantum statistic, sufficiency, von Neumann algebra.

Introduction

In the paper we investigate the notion of sufficiency in quantum statistics. Due to the richness of the noncommutative (-= quantum) setup there are several (nonequivalent!) possible definitions of this term. The most popular ones describe sufficiency as a property of subalgebras of a given algebra without referring to a particular quantum statistic. We adopt the approach which is closely related to the classical factorization criterion, and emphasize the role of the statistic itself rather than the underlying subalgebra. This approach seems to be especially well motivated when we are dealing with the full algebra of bounded operators on a Hilbert space together with vector states, which is just the case we are interested in.

The main results of the paper are contained in Sections 3 and 4, where we find conditions characterizing the weak sufficiency of a given quantum statistic. In Section 3 we obtain a general answer to this problem, while Section 4 provides a simpler and more straightforward solution in the case where our statistic is discrete. Sections 1 and 2 are devoted to introducing the reader, especially not familiar with (or having little knowledge of) the subject, to some basic notions and ideas of quantum statistics.

1. Preliminaries, notation and the quantum setup

Let 7-/ be a Hilbert space. By 79(7-/) we shall denote the lattice of all orthogonal projections acting in 7-/. For p ~ 79(7-/) we set p ± = 1 - p , where 1 is the identity operator on 7-/. For a closed subspace /C of 7-/ the orthogonal projection onto /C

[367]

368 K. LUBNAUER, A. LUCZAK and H. PODSI~DKOWSKA

will be denoted by Proj K~. B(7-() will stand for the algebra of all bounded linear operators on ~ .

By a von Neumann algebra .M of operators acting on 7/ we mean a ,-algebra .M c B(~) which is closed in the strong operator topology on ]~(7-/), i.e. the topology given by the family of seminorms

~ ( ~ ) ~ x ~ IIx& II, ~ • 7-t,

We shall not use any advanced theory of von Neumann algebras; some basic necessary facts can be found for instance in [4, 5, 9].

A vector ~ e 7-/ is said to be cyclic (generating) for a v o n Neumann algebra .M if

M ~ = {x~ : x • M } = ~ .

is said to be separating for .M if for each x e .M the equality x~ = 0 implies x----0.

The a-field of Borel subsets of the real line ~ will be denoted by /3(~). The following simple lemma which will be needed in the sequel is a part of

the folklore of Hilbert space theory. We omit its standard proof.

LEMMA 1. Let ~ be a separable Hilbert space and let {~o0 : 0 e ®} C 7-/. Then there is a countable subset {~oon : n = 1, 2 . . . . } o f {~o0 : 0 e ®} such that

[{qg0n : n = 1,2 . . . . }] = [{q90 : 0 e 0}],

where for /~ C 7-/, [E] stands for the smallest closed linear subspace of containing IC. []

The most basic 'probability space' employed to describe a quantum system consists of a separable Hilbert space 7-f and the so-called pure (or vector) state represented by a unit vector ~0 e 7-/. A 'noncommutative (or quantum) random variable' called usually observable is a self-adjoint operator T on 7-/, so that for the spectral decomposition

F T = ~. e(d~.) (1)

of T, where e : B ( I R ) ~ "P(~) is the spectral ( - projection-valued) measure of T, the quantity

(e(g)qg, 99) = Ile(E)~0112, E • /3(R) ,

represents the probability that being in the state q9 the observable T takes value in the set E.

The function #~0(-) = (e(.)~0, ~0) is a genuine probability distribution on the space (I~, /3(R)) of all possible values of the observable T, and in quantum statistics we are dealing with a family {~00 : 0 • ®} of possible states from which we want to pick up the true state of our physical system. Consequently, a 'quantum statistical space' in our case is (~ , {qg0:0 • ®}). As in the classical case, various observables (i.e. random variables) prove useful in obtaining information about this true state, so we shall use for them the name of (quantum) statistics.

WEAK SUFFICIENCY OF QUANTUM STATISTICS 369

2. Sufficient quantum statistics

The first idea of sufficiency in the noncommutative setting goes back to H. Umegaki (see [10, 11] for the original definition and [2] for its refinement) and can in our context be described as follows. Let .A/" be the yon Neumann algebra generated by a quantum statistic T. Then for T having the spectral decomposition given by Eq. (1) we have

I A/" = c~(T) = c~()~) e(d)~) • (3O

- - a complex-valued bounded Borel function]. !

Thus .N" is an abelian von Neumann algebra determined by spectral measure e. T is said to be sufficient in Umegaki's sense for a family of states {~00 : 0 E ®} if there exists a normal conditional expectation E : B(7-{) ~ .A/" such that for any 0 E ® and x E ]~(H) we have

(E(x)~00, ~00) = (x~oo, ~oo),

i.e. the states ~00 are E-invariant. This definition of sufficiency is best suited to the case when we have an arbitrary von Neumann algebra A/[ instead of B(H), an arbitrary (not necessarily abelian) von Neumann subalgebra A/" of A//, and an arbitrary family of (not necessarily vector) states {~P0 : 0 E ®}, in which case we speak of the sufficiency of the algebra A/'. The requirement that the map E : .34 -+ .N" is a normal conditional expectation was afterwards replaced in the papers [3, 6, 7] by a weaker condition that E is completely (or two)-positive; nevertheless, the very idea of sufficiency as the possibility of state-invariantly mapping an algebra into its subalgebra remained intact. It is obvious that if the subalgebra .M is arbitrary (nonabelian) it does not come from a self-adjoint operator in the way described above, thus we cannot speak about the sufficiency of a quantum statistic. Moreover, in our case when we are dealing with the full algebra B(H), the requirement of the existence of a normal conditional expectation onto A/" puts strong restrictions on A/'; in particular, it must be atomic (cf. [8, Section 10.21]), which means that the statistic T has the discrete spectral decomposition

T = ~ )~kek. k

Thus adopting Umegaki's definition would lead to a considerably narrower class of statistics. However, there is another natural way of treating a given statistic as sufficient for a family of vector states {qg0 : 0 E ®}. Namely, the states qg0 can be obtained from some state to which we apply statistics being functions of our initial one. In the classical case this is given by the well-known factorization criterion which says that if {Po : 0 E ®} is a family of distributions dominated by a a-finite measure # with densities fo with respect to #, then a statistic T is sufficient for this family if and only if there exist real-valued Borel functions ~0 and a Borel


function h such that

fo = O o ( T ) h, Po - almost everywhere.

In the quantum case a natural generalisation of the factorization criterion has been proposed by E. Czkwianianc in [1] where a statistic T is called sufficient for a family of states {~00 " 0 6 (9} if there exist Borel functions (I)0 " R --+ R and a unit vector X in 7-[ such that

qgo = O o ( T ) x for all 0 ~ (9.

This definition needs a slight refinement for the following simple reason. Namely, if q9 is a state of a system, then for each complex number c of modulus one cop represents the same state. Indeed, we have

#c¢( ' ) = (e(.)c~o, c~o) = [cl2(e(.)~o, 99)

= (e(-)(p, (p) = / ~ ( - ) ,

thus for sufficiency we would only require the equality

(P0(T)x = coqgo,

with some complex numbers co of modulus one. To put it in a simple way, let us agree to call a vector ~" a version of the vector ~0 if there is a complex number c of modulus one such that

~ ' = c~o.

We adopt the following definition.

DEFINITION. A statistic T is weakly sufficient for a family of states {~o0 " 0 6 ®} if there exist Borel functions (P0 " R ~ IR, a unit vector X in ~ , and a version ~'0 of ~00 for each 0 ~ (9, such that

~o = O o ( T ) x for all 0 ~ (9. (2)

Obviously, the requirement that IIx II = 1 is inessential, and we shall omit it in the sequel.

Thus in this approach the weak sufficiency of T, or equivalently of the algebra A/" generated by T, consists in the possibility of obtaining all the states ¢P0 by means of some observables which are functions of T. We have chosen the name 'weakly sufficient' in order to distinguish this notion of sufficiency from the above-mentioned notions due to H. Umegaki; E Hiai, M. Ohya, M. Tsukada; and D. Petz. Moreover, it can be shown that weak sufficiency follows from sufficiency, but not the other way round, which additionally justifies its name.

Let us comment here on one important point. Since weak sufficiency is weaker than any of the notions of sufficiency considered so far, it is natural to ask whether some 'classical' properties of sufficiency carry over to this new setting. A more detailed answer to this question requires deeper analysis of weak sufficiency. Here, by way of example, we would like to indicate only two points. First, it turns out that the notion of minimality of a weakly sufficient statistic can be introduced in a natural


way, and a criterion of minimality can be obtained, at least for discrete statistics. Second, from the classical Blackwell-Rao theorem the following fundamental fact follows. Let S be a (classical, commutative) sufficient statistic which is an unbiased estimator of a function g : ® --+ ]K. For each unbiased estimator T of the function g there is a function cI) such that q~(S) is an unbiased estimator of g, and the variance of qb(S) is less than or equal to the variance of T. However, this is not the case for weakly sufficient statistics. Namely, according to [1, Example 2.2] there is a weakly sufficient statistic S with the following properties:

(i) S is the only unbiased estimator of the parameter 0 in the von Neumann algebra generated by S.

(ii) The variance of S is large when compared with the variance of some other unbiased estimator of 0.

This shows that if we consider functions of a weakly sufficient statistic which are unbiased estimators of some function of the parameter 0 we cannot in general expect decreasing of the variance.

Now a question arises about the existence of a weakly sufficient statistic for a given family of states. The following result was essentially obtained in [1].

THEOREM 1 (cf. [1, Theorem 1.5]). Let {~on : n = 1, 2 . . . . } be any sequence of states. There exists a weakly sufficient statistic for this sequence if and only if for each n there is a version ~n of qgn such that

('~i, ~j) E ~ for all i, j = 1, 2 . . . . .

3. The main problem

Now we want to address the following problem: given a family {~00 : 0 6 ®} of states and statistic T, find conditions characterizing the weak sufficiency of T for {~P0}. Before giving a solution to this problem, let us introduce some necessary notions.

For a von Neumann algebra of operators .A/[ acting on a Hilbert space ~ we denote by .A41 the commutant of .A4, i.e. the algebra of all bounded operators on 7-t which commute with all the operators from A4. In particular, if A4 is abelian then A4 C M I.

An abelian von Neumann algebra AA is called maximal abelian if .A4 = AA'. Let p' be a projection in AA'. Then we can consider the operators xp' with

x c .A4 restricted to the space p ' ~ . The von Neumann algebra

M p , = { x p l l p ' ~ : x E All}

of operators acting on the Hilbert space p t ~ is called the induced von Neumann algebra. Let us observe that the map

A/[ ~ x ~ xp'[p'7-( c Adp,

is a normal ,-homomorphism between the algebras A4 and .A4p,. Since such homomorphisms take the projections of one algebra onto the projections of the


other, the following simple lemma follows. Nevertheless, to make the exposition selfcontained, and to restrict the use of the theory of von Neumann algebras to a minimum, we give its proof here.

LEMMA 2. Let .All be a von Neumann algebra acting in a Hilbert space ~ , and let p' be a projection in M I. For each projection q in ./k4p, there is a projection f in .All such that q = fpllp'7-[.

Proof: We have q = xp'Jp'7-[

for some x 6 .A/[. It follows that xp' is a projection on ~ , so we have

xp' = (xp') 2 = x2p '.

Let r(x) be the range projection of x. Then r(x) E A4, and

r(x)x = x; (3)

moreover, r(x) is the minimal projection satisfying (3). r(x)p' is also a projection, and

r(x)p'xp' = r(x)xp' = xp',

which means that xp' <<, r(x)p'.

Furthermore, x p ' + p'± is a projection too, and

(xp I + p1±)x = x2p I + xp t± = xp I + xp '± = X,

thus the minimality of r(x) yields

r(x) <~ xp I + p1±.

Multiplying both sides of the above equality by p' on the right and on the left we obtain

plr(x)pl = r(x)p I <~ p'xp I = xp',

showing that x p ' = r(x)p', which finishes the proof. []

MAIN THEOREM (general case). Let T be a quantum statistic on a separable Hilbert space 7-[ with the spectral decomposition given by (1), and let {~Oo :0 ~ 6)} be a family of vector states in 7-[. Denote, as before, by Af the von Neumann algebra generated by T. The following conditions are equivalent:

(i) T is weakly sufficient for {~Oo : 0 ~ 6)}. (ii) There is a projection p' in A~ "I such that {~oo : 0 ~ 6)} C p ' ~ and the induced

algebra A/'p, is maximal abelian, and for each 0 ~ 6) there is a version ~o of ~Oo such that for any Borel set E C ~ and any 0 I, 0 I' c 6) we have

(e(E)~o,, ~o") ~ R. (4)


Proof: Assume that T is weakly sufficient for {q)0}, and let ~'0 be a version of ~P0 such that (2) holds with some Borel functions ~ 0 " R ~ R, and X ~ 7-/. For each 0 6 ® there is a sequence {a~ ")} of self-adjoint operators in N" such that

~o = ~o(T)x = lim a~n) X in norm, n ----~ o o

which implies that ~0 (and thus q)0) belongs to N'X. Denote by p ' the projection onto N'X. Then p! 6 N" since the subspace N'X is N'-invariant (cf. [4, Section 2.6]), ~o0 ~ N'X = p ' ~ , for each 0 6 ®, and since X is a cyclic vector for the abelian algebra N'p,, it follows that Np! is maximal abelian (cf. [5, Corollary 7.2.16]). Furthermore, for each E 6 B(R) and any 0 !, 0 " 6 ® we have

- (n) v (m) (e(E)~o,, ~0") = lim (e(E)a o, ^,ao,, X) n ,m- -+oo

= lim (a(~m)e(E)a(f) X, X) ~ N, n , m --+ oo t~ u

(m) . . r . . , (n) since ao,, etr.)a o, is self-adjoint.

Now assume that condition (ii) holds. Let {~'0, : n = 1, 2 . . . . } be a countable subset of {~'0:0 6 ®} such that

[{~'0. : n = 1, 2 . . . . }1 = [{~0 : 0 6 ®}1. (5)

To simplify the notation, denote =

Restrict attention to the space p ' ~ . We define inductively a sequence {Xn'} of vectors in p'7-(, and sequences {Pn}, {qn} of projections in N'p, as follows. Put

X~ = ~ 1 , "]-'~1 =N'p ,X~, Pl = P r o j ~ l , q, = p , .

Having defined Xnt 1 , P . - I and q , - l , we put

X~. = q.~- l ~. , 7-(. = "/V'p' X~. ,

We have !

Xn E '~"~n,

Put

p . = IProj 7-(n,

7-(i-LT"gj for i 7~j,

1 ! x' = F , - ; x . ,

n = l

so that

q . = P l + ' " + P . .

pn E./V;, = N'p,.

X ! n npnX !"

Since ~n = q n - l q ) " n at" q~-,~. = qn-l~. + X~,

it follows that ~'n c ~ l (9 "'" (9 ~ . . Decompose ~'. as

= % + ' " + n where ~ k , k = l . . . . . n.

374 K. L U B N A U E R , A. L U C Z A K and H. P O D S t ~ D K O W S K A

Each ~(k) ~u n can be arbitrarily closely approximated by an element a~xk with some ak • .Me,, which means that ~ can be arbitrarily closely approximated by the sum

n n

Z ak)(~ = Z ak (kpk) X' = a X', k = l k = l

where n

a = Z kakpk E A/p,. k = l

It follows that ~'n • A/p,X' for each n. Consequently, Eq. (5) yields the inclusion

: 0 • e } c A/p,x'. Let q be the projection onto A/p,X'. Since the subspace A/p'X' is A/p,-invariant,

it follows that q • A/p, = A/p, (cf. [4, Section 2.6]). For arbitrary Olk • R and Ek • / 3 ( ~ ) we have on account of (4)

m m

so because each self-adjoint a • A/ is a limit of the sums Y'~kml ake(Ek) in the strong operator topology, we get for each 0', 0 " 6 ®

(a~o,, ~o") • ~. (6)

Furthermore, for each E •/3(]~) and each 0 • (9

x ' / = e(E)go, nXn = q = n = l n

Since the q_Ln_l are projections in A/p,, from Lemma 2 we obtain

qL1 = enp'lP '7(

for some projections en in A/. Consequently, condition (6) gives

°° 1 £ 1 (e(E)~o, X') = E -(e(E)~o, enp'~n) ---- -(e(E)go, en~n)

n = l n n = l /'/

=- £ l(ene(E)~o, ~n) • ]~x, n=l n

since ene(E) is a self-adjoint element of A/. It follows that (e(.)~0, X') is a real- valued signed bounded measure, clearly it is absolutely continuous with respect to the measure (e(-))(, X') = Ile(')x'll 2- Let ~0 be a density of (e(.)~'0, X') with respect to (e(.)X', X'), i.e. ~0 is a real-valued Borel function such that

(e(E)~o, X') = [ ~o(X) (e(d•)x', X'), de


for each E • /3(R) . Let

E ° = {~. • ~ : [qb0()Q[ ~< n}.

For each E • /3 (R) we have

o _ fE ~oOQ <e(d)Qx', X'). (e(E)e(E°)~o, X') = (e(E fq E,)cpo, X') = nE °

Using operator-valued integrals we can write

(e(E)e(E°)~o,X ') =<fE*o(X)IEo(X)e(dX)x',x'>, (7)

where 1E stands for the indicator of the set E. Taking into account that the operator • o(T)e(E°n) belongs to A/, Eq. (7) can further be rewritten in the form

< F > (e(E°)~o, e(E)x') = e(E) *o()~) e(d)~)e(E°)x ', X' O0

= ( f _ ~ o ( ) Q e ( d , ~ ) e ( E ° ) x ' , e ( E ) x ' > = (~o(T)e(E°)x ' ,e (E)x ' ) .

Thus for an arbitrary linear combination ~ - - 1 olke(Ek) we have

m m

<e(E°)~o, Zotke(Ek)X ' ) = <*o(T)e(E°)x ', Zo tke (Ek)x ' ) , k=l k=l

and since such combinations approximate each a • A/ we get

(e(E°)~o, ax') = (*o(T)e(E°)x ', ax'). (8)

Now )(' • p t~ , thus

• o (T )e (E° )x '= *o(T)e(E°n)p'x '= p'*o(T)e(E°n)X ',

so ~ o ( T ) e ( E ° ) x ' • p'7-[; moreover Eq. (8) takes the form

(e( EO)~o, ap,x, ) o , , , = <~o(T)e(E,,)X, ap X ),

which means that <e( E°)~o, zX') o , = (~o(T)e(E, ,)X, zX') (9)

for each z • A/p,. Taking into account that zX' • A/p'Z' = q~ , we can write Eq. (9) in the form

0 (e(En)qgo, ZX') = (~o(T)e(E°n)X ', qzx'> (10)

= <qOo(T)e(E°n)X ', zx'>,

where the last equality is justified by the fact that ~o(T)e(E°n)X ' • p '~ , and thus

we may apply to ' the projection q • A/p,. Since e(E°n)~O • A/p'X', we get from (10)

e(E°)~o = q~o(T)e(E°n)X '.


Since q is a projection in A/p,, Lemma 2 yields I ! q = fp [pT-[,

for some projection f in A/. Consequently,

e(E°n)~O = fp'~o(T)e(E°n)X'= f~o(T)e(E°,,)X'= ~o(T)e(E°n)(fx').

Putting

we obtain

N o w ,

and

X = f x ' ,

e(E°)~o = ~o(T)e(E°n)X.

lim e(E°n)X = X,

lim ~o(T)e(E°)x = lim e(E°)~o = ~o~, n ---~ o ~ n---+ o o

which on account on the closedness of eoo(T) means that X is in the domain of • o(T), and

= o(T)x,

finishing the proof. []

To clarify the somewhat mysterious nature of the requirement that the algebra A/p, is maximal abelian, and at the same time to show its essentiality, consider the following example.

Let the algebra A/ generated by the statistic T be maximal abelian, and let ~p be a cyclic separating vector for A/. Take an arbitrary projection p a .N" such that O ~ p ~ l , and put

Ol- IIp lt' o2- llp±7,1 .

Then it is easily seen that T is weakly sufficient for the family {~Ol, ~o2}, since we can take p' = 1, so that A/p, = A/, and

(e(E)~ol, ~o2) = 0,

so the same equality holds for each version ~'1, ~'2 of ~o1, ~o2. Now take

[ : o. l T~--~ .

T is a quantum statisticin the Hilbert space H @ ~ . Denote by . ~ the yon Neumann algebra generated by T. Then if T has the spectral measure e it is clear that T has the spectral measure ~ of the form

e ' ( E ) = Ie(E) e(E)O ] , E~B(II~),

WEAK SUFFICIENCY OF QUANTUM STATISTICS

consequently, for each complex-valued bounded Borel function do we have

o 1 do(T) = do(T)

thus

Put

• d o - - a complex-valued bounded Borel function ] .

~01 = x/llpTtll2 + ]]p±~n 2 p±7 r , x/Hp~pn2+ ]]p±~l] 2 \ p~P J

377

We have

~'(E)~I, ~2) = ((e(E)p~r, p ± ~ ) + (e(E)p±~t, p~r)) = O, jlp~p]12 + ][p±~]]2

so equality (4) in condition (ii) of the Main Theorem is satisfied. Assume that is weakly sufficient for {~j, ~'2}. Then there exist a vector

in ~ @ ~ and real-valued Borel functions dol, do2 such that

dol(T) " = c1 '1, do2(T)2 = c2 '2, (11)

for some complex numbers cl, c2 of modulus one. Eqs. (11) can be rewritten as

doI(T)x (j) = c l p ~ , dol(T)x ~2) = c l p ± ~ ,

do2(T)x (1) ~-- c2p±lP, do2(T)x (2) ---- c2p~.

Then we have

[gldol(T) + ?2do2(T)]x O) = glclp~P + C'2c2p±~P = ~P,

which implies that X (J) is separating for .A/'. Indeed, if a x (1~ = 0 for a 6 iV', then

a ~ = a[cl dol(T) + c2doz(T)]x (1) -- [Cl dol (T) + c'2doz(T)]a X (1) _- 0,

thus a = 0, because ~ is separating for .A/'. We have further

pdoz(T)x ~'l) = p ( c z p ± ~ ) = O,

consequently pdo2(T) = O,


which yields the equality

0 = p~2 (T )x (2~ = p(c2pgr) = c2pge,

and so p ~ = 0, which means that p = 0, a contradiction. Thus T is not weakly sufficient for {~'1, ~e}.

4. The discrete case

The conditions of weak sufficiency in the general case obtained in the previous theorem are a little complicated. It turns out that they can be given a simpler form in the case where our statistic has a discrete spectral decomposition. Namely, we have the following theorem.

MAIN THEOREM (discrete case). Let T be a quantum statistic on a separable Hilbert space 7-[ with the spectral decomposition

T = Z ) ~ k e k ' (12)

where {ek} is a countable partition of the identity, and )~k • IR are different eigenvalues of T, and let {q)0:0 • 19} be a family of vector states in 7-[. The following conditions are equivalent:

(i) T is weakly sufficient for {q)0 : 0 • (9}. (ii) For each k, dim[{ek~oo : 0 • (9}] = 0 or 1, and for each 0 • (9 there is

a version ~o of~oo such that for each k and any 0',0 'I • ®, we have (ek~o', ~o") • IR.

Proof: Assume first that T is weakly sufficient. Then for some unit vector X • ~ , Borel real-valued functions q%, and for all 0 • (9 we have for some version ~'o of ~Po 0 = *o(T)x = Z *O()~k)ekx = Z a k e k X , (13)

k k

where oe ° = qb0 (Xk).

Fix k. Applying ek to both sides of (13), we get for each 0 • (9,

e k ~ O 0 O t k e k X ,

which means that the linear space spanned by {eke0 : 0 • ®} (and thus the linear space spanned by {ek~oo : 0 • (9}) has dimension 0 or 1 (it is spanned by the vector ekX). Moreover, for arbitrary 0 t, 0 " • (9 we have

0 t 0 ~ (ek~o,, ~o") = (ek~o', ek~o,,) = (ak e~x, ot k ekx2

0 t 0 tr = ~ ~k Ilekx I12 • R.

Assume now that condition (ii) holds, and let ~0 be a version of ~o0 as in condition (ii). For each k such that dim[{ek~oo : 0 • (9}] = 1 choose Ok for which

ek~ok 5 0 ,


and put

For each 0 e (9

with some ot °. We have

X = Z #ekcPOk. k

ek~o ---- otOekqg~ok

[[ek~o [[2 ~_ (oeOek~ok, ek~o) = ot ° (ek~o~, ek~o)

= or° (ek~ok, ~o),

which means that a ° e R. Define the functions qbo by

[kod, if ~. X~ ~0 (~-) = [ 0, otherwise,

where the Xk are as in (12). Then the ~0 are Borel real-valued functions, and

c~°(T)x = Z *o(XIc)ekx = Z kot°~l #ekqgOk k k

= e k = q)O,

k

showing that T is weakly sufficient. []

The proof of the discrete case above is straightforward and independent of the general case. Using the Main Theorem (general case) we also have another proof.

Second proof: In order to find out that the discrete case is a particular case of the general one, it is enough to show that the condition

dim[{ek~po : 0 ~ ®}] = 0 or 1 (14)

for each k, is equivalent to the existence of a projection p' in N" such that {~00:0 e ®} C p'7-L and the algebra Np, is maximal abelian.

Thus let us assume that condition (14) holds, and let p~ be the projection onto [{eke00" 0 6 ®}]. Then p~ ~< ek, consequently p~ c N" because p~ commutes with all ei. Moreover, the p~ are one- (or zero)-dimensional. Put

p ' = Z p ;. k

Then p' 6 N", and the algebra N'p, is generated by the spectral measure {p~}. Since the p ' k are minimal projections it follows that N'p, is maximal abelian (cf. [5, Theorem 9.4.1]). For each 0 6 ® we have

q)O ~ Z t pk ~oo = pl~po ,

and thus ~P0 e p'7-/.


Conversely, if p' is a projection in Af such that {~o0 " 0 6 ®} C p'7-/, and the algebra .Alp, is maximal abelian, then since N'p, is generated by the spectral measure {p 'ek} it follows that the p 'ek are minimal (i.e. zero or one-dimensional) projections (cf. [5, Theorem 9.4.1]), and for each k we have

{ekqgO " 0 E ®} = {ekp'~oO " 0 ~ ®}

= {p'ekqgo " 0 E ®},

which means that condition (14) holds. []

Acknowledgements The authors are indebted to the referee for pointing out and correcting an

inaccuracy in the proof of the Main Theorem, as well as for many valuable remarks concerning the previous version of the paper.

REFERENCES

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Weak sufficiency of quantum statistics

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